Dynamics of a plant–herbivore model with differential–difference equations

References

• Agiza, H. N., ELabbasy, E. M., EL-Metwally, H., & Elsadany, A. A. (2009). Chaotic dynamics of a discrete prey-predator model with Holling type II. Nonlinear Analysis Real World Applications, 10, 116–129.10.1016/j.nonrwa.2007.08.029
   Web of Science ®Google Scholar
• Busenberg, S., & Cooke, K. L. (1982). Models of vertically transmitted diseases with sequential continuous dynamics. Nonlinear phenomena in mathematical sciences. New York, NY: Academic Press.
   Google Scholar
• Caughley, G., & Lawton, J. H. (1981). Plant-herbivore systems. In R. M. May (Ed.), Theoretical ecology (pp. 132–166). Sunderland: Sinauer Associates.
   Google Scholar
• Chattopadhayay, J., Sarkar, R., Frıtzsche-Hoballah, M. E. F., Turlıngs, T. C. J., & Bersıer, L. F. (2001). Parasitoids may determine plant fitness—A mathematical model based on experimental data. Journal of Theoretical Biology, 212, 295–302.10.1006/jtbi.2001.2374
   PubMed Web of Science ®Google Scholar
• Cooke, K. L., & Györi, I. (1994). Numerical approximation of the solutions of delay-differential equations on an infinite interval using piecewise constant argument. Computers & Mathematics with Applications, 28, 81–92.
   Web of Science ®Google Scholar
• Danca, M., Codreanu, S., & Bakó, B. (1997). Detailed analysis of a nonlinear prey-predator model. Journal of Biological Physics, 23, 11–20.10.1023/A:1004918920121
   PubMed Web of Science ®Google Scholar
• Das, K., & Sarkar, A. K. (2001). Stability and oscillations of an autotroph-herbivore model with time delay. International Journal of Systems Science, 32, 585–590.10.1080/00207720117706
   Web of Science ®Google Scholar
• Edelstein-Keshet, L. E. (1986). Mathematical theory for plant-herbivore systems. Journal of Mathematical Biology, 24, 25–58.10.1007/BF00275719
   Web of Science ®Google Scholar
• Feng, Z., Qiu, Z., Liu, R., & DeAngelis, D. L. (2011). Dynamics of a plant–herbivore–predator system with plant-toxicity. Mathematical Biosciences, 229, 190–204.10.1016/j.mbs.2010.12.005
   PubMed Web of Science ®Google Scholar
• Gopalsamy, K., & Liu, P. (1998). Persistence and global stability in a population model. Journal of Mathematical Analysis and Applications, 224, 59–80.10.1006/jmaa.1998.5984
   Web of Science ®Google Scholar
• Gurcan, F., Kartal, S., Ozturk, I., & Bozkurt, F. (2014). Stability and bifurcation analysis of a mathematical model for tumor-immune interaction with piecewise constant arguments of delay. Chaos Solitons & Fractals, 68, 169–179.
   Web of Science ®Google Scholar
• Hone, A. N. W., Irle, M. V., & Thurura, G. W. (2010). On the Neimark–Sacker bifurcation in a discrete predator-prey system. Journal of Biological Dynamics, 4, 594–606.10.1080/17513750903528192
   PubMedGoogle Scholar
• Kartal, S., & Gurcan, F. (2015). Stability and bifurcations analysis of a competition model with piecewise constant arguments. Mathematical Methods in the Applied Sciences, 38, 1855–1866.10.1002/mma.v38.9
   Web of Science ®Google Scholar
• Lebon, A., Mailleret, L., Dumont, Y., & Grognard, F. (2014). Direct and apparent compensation in plant-herbivore interactions. Ecological Modelling, 290, 192–203.10.1016/j.ecolmodel.2014.02.020
   Web of Science ®Google Scholar
• Li, Y. (2011). Toxicity impact on a plant-herbivore model with disease in herbivores. Computers Mathematics with Applications, 62, 2671–2680.10.1016/j.camwa.2011.08.012
   Web of Science ®Google Scholar
• Liu, P., & Gopalsamy, K. (1999). Global stability and chaos in a population model with piecewise constant arguments. Applied Mathematics and Computation, 101, 63–88.10.1016/S0096-3003(98)00037-X
   Web of Science ®Google Scholar
• Liu, R., Feng, Z., Zhu, H., & DeAngelis, D. L. (2008). Bifurcation analysis of a plant–herbivore model with toxin-determined functional response. Journal of Differential Equations, 245, 442–467.10.1016/j.jde.2007.10.034
   Web of Science ®Google Scholar
• May, R. M. (2001). Stability and complexity in model ecosystems. Princeton, NJ: Princeton University Press.
   Google Scholar
• Mukherjee, D., Das, P., & Kesh, D. (2011). Dynamics of a plant-herbivore model with Holling type II functional response. Computational and Mathematical Biology, 2, 1–9.
   Google Scholar
• Ortega-Cejas, V. O., Fort, J., & Méndez, V. (2004). The role of the delay time in the modeling of biological range expansions. Ecology, 85, 258–264.10.1890/02-0606
   Web of Science ®Google Scholar
• Owen-Smith, N. O. (2002). A metaphysiological modelling approach to stability in herbivore–vegetation systems. Ecological Modelling, 149, 153–178.10.1016/S0304-3800(01)00521-X
   Web of Science ®Google Scholar
• Öztürk, I., Bozkurt, F., & Gurcan, F. (2012). Stability analysis of a mathematical model in a microcosm with piecewise constant arguments. Mathematical Biosciences, 240, 85–91.10.1016/j.mbs.2012.08.003
   PubMed Web of Science ®Google Scholar
• Saha, T., & Bandyopadhyay (2005). M. Dynamical analysis of a plant-herbivore model: Bifurcation and global stability. Journal of Applied Mathematics & Computing, 19, 327–344.
   Google Scholar
• Sui, G., Fan, M., Loladze, I., & Kuang, Y. (2007). The dynamics of a stoichiometric plant-herbivore model and its discrete analog. Mathematical Biosciences Engineering, 4, 29–46.
   PubMed Web of Science ®Google Scholar
• Sun, G. Q., Chakraborty, A., Liu, Q. X., Jin, Z., & Anderson, K. E. (2014). Influence of time delay and nonlinear diffusion on herbivore outbreak. Communications in Nonlinear Science and Numerical Simulation, 19, 1507–1518.10.1016/j.cnsns.2013.09.016
   Web of Science ®Google Scholar
• Zhao, Y., Feng, Z., Zheng, Y., & Cen, X. (2015). Existence of limit cycles and homoclinic bifurcation in a plant-herbivore model with toxin-determined functional response. Journal of Differential Equations, 258, 2847–2872.10.1016/j.jde.2014.12.029
   Web of Science ®Google Scholar